Page 45 - Introduction to Computational Fluid Dynamics

P1: IWV/ICD
10:49
May 25, 2005
CB908/Date
0521853265c02
24
where 0 521 85326 5 1D HEAT CONDUCTION

kA
AE = (2.25)
x
e

kA
AW = (2.26)
x w
,n ,o
S = ψ q + (1 − ψ)q V
P P
o o
+ (1 − ψ) AE T + AW T
E W
o

ρ VC o
+ − (1 − ψ)(AE + AW) T . (2.27)
P
t
P
Note that Equation 2.24 has the same form as Equation 2.15, but there are
important differences:
1. Coefficients AE and AW can never be negative since kA/ x can only assume
positive values.
2. AE and AW are also amenable to physical interpretation; they represent
conductances.
3. Again,insteady-stateproblems,ψ = 1because t =∞.Inunsteadyproblems,
o
for certain choices of t, however, the multiplier of T can still be negative.
P
This observation is in common with the TSE method.
2.5 Stability and Convergence

Before discussing the issues of stability and convergence, we recognize that there
will be one equation of the type (2.24) [or (2.15)] for each node P. To minimize
writing, we designate each node by a running index i = 1, 2, 3,..., N, where i = 1
and i = N are boundary nodes. Thus, Equations 2.24 are written as

AP i T i = ψ [ AE i T i+1 + AW i T i−1 ] + S i , i = 2, 3,..., N − 1, (2.28)

where superscript n is now dropped for convenience. In these equations, AP i
represents multiplier of T P in Equation 2.24.
It will be shown later that this equation set can be written in a matrix form
[A][T] = [S], where [A] is the coefficient matrix and [T] and [S] are column vectors.
This set can be solved by a variety of direct and iterative methods. The methods
yield converged solutions only when the condition for convergence (also known as
Scarborough’s criterion [64]) is satisfied. To put it simply, the criterion states that

Condition for Convergence
ψ [|AE i |+|AW i |]
≤ 1 for all nodes, (2.29)
|AP i |